What is the integral of 5x(sin(x)) ?

The integral of 5x(sin(x)) is 5(-xcos(x)+sin(x))+C

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integral of 5x(sin(x))

\int 5 x (sin (x)) d x

5 (- x cos (x) + sin (x)) + C

Solution steps

\int 5 x sin (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int x sin (x) d x

Apply Integration By Parts

: - x cos (x) - \int - cos (x) d x

= 5 (- x cos (x) - \int - cos (x) d x)

\int - cos (x) d x = - sin (x)

= 5 (- x cos (x) - (- sin (x)))

Simplify

= 5 (- x cos (x) + sin (x))

Add a constant to the solution

= 5 (- x cos (x) + sin (x)) + C

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\frac{d y}{d x} + 4 y = 11

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What is the integral of 5x(sin(x)) ?
The integral of 5x(sin(x)) is 5(-xcos(x)+sin(x))+C